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Size effect on deformation of duralumin micropillars – A dislocation dynamics study
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ScienceDirect Scripta Materialia 76 (2014) 73–76 www.elsevier.com/locate/scriptamat
Size effect on deformation of duralumin micropillars – A dislocation dynamics study R. Gu,⇑ P.P.S. Leung and A.H.W. Ngan Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China Received 7 November 2013; accepted 26 December 2013 Available online 3 January 2014
Two-dimensional dislocation dynamics simulations are used to study precipitate strengthening effects in duralumin micropillars. The results show that a refined microstructure may resist and slow down the movement of dislocations inside the confined volume, leading to hardening and a weak dependence of strength on size. This study illustrates that the deformation behavior of small crystals is controlled by the combined effect of the internal length scale and the external size of the crystal. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Plastic deformation; Dislocation dynamics; Size effect; Precipitate hardening
The effect of size on the deformation behavior of metallic materials has been intensively studied in recent years. One of the most significant findings is that metallic single crystals show a power-law size dependence of strength [1–3] and jerky stress–strain behavior in a stochastic manner [1,4–9]. With evidences gained from in situ transmission electron microscopy [9,10] and Laue microdiffraction [11,12], a number of physical models have been devised to explain the deformation behavior of small crystals. The “dislocation starvation” [6,13] theory considers that mobile dislocations in a small crystal would easily slip through the small crystal without accumulating and multiplying, so that the crystal is constantly maintained in a dislocation-free state. Another group of models consider that the small crystal is not dislocation free initially, but the operation of dislocation sources [14,15] and the interaction of dislocations in a confined environment lacking the mean field-averaging effect [16,17] may result in unusual deformation behavior. Another model [18] explicitly predicts that a power-law size dependence of strength stems from Taylor-type interactions in a fractal dislocation network, and that the power-law exponent is inversely related to the fractal dimension of the dislocation network. In addition to the extrinsic size effect, several recent studies have explored how the mechanical properties of small crystals are affected by changes in the intrinsic
⇑ Corresponding author. Tel.: +852 28592624. e-mail: raicy4197@ gmail.com
microstructural length scale, such as decreasing the interaction distances between dislocations by trapping them inside the crystal by surface coating [19–22], or introducing precipitates [23], grain boundaries [24,25], or nanocrystalline [26,27] or nanotwinning [22,28,29] microstructures into the crystal. Our previous experimental study [23] on the deformation behavior of duralumin micropillars has shown that, with the presence of second-phase precipitates, strength is much improved and less size-dependent, and a very high density of dislocation debris is retained in the deformed microstructure, suggesting that mobile dislocations are slowed down by viscous drag from the precipitates. In this study, two-dimensional (2-D) dislocation dynamics simulations [30–33] are used to study the deformation behavior of such duralumin micropillars. The distribution of the precipitate particles and the external size of the specimen are varied to investigate their effects on the plastic properties of the micropillars. In the present 2-D simulations, screw dislocations are considered, and these are randomly distributed in a simulation region with a given initial density q0. The simulation region is rectangular, with the short and long sides (in a ratio of 1:3) respectively representing surfaces parallel and perpendicular to the radial direction of the micropillar. Periodic boundary conditions are applied at the two short sides of the simulation region, meaning that as a dislocation escapes from one side it re-enters into the region from the opposite side. The other two (long) sides are regarded as free surfaces where nearby dislocations would be affected by image forces and
1359-6462/$ - see front matter Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.scriptamat.2013.12.020
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may therefore likely be depleted on them. Two orthogonal slip systems are considered in the simulation region, which are oriented at 35.2 and 125.2° from the radial direction. The material properties of aluminum are used in this simulation. Dislocations gliding in the slip plane may sustain three types of forces without meeting any precipitates: (i) the pairwise elastic interaction force lb2 between dislocations, where rij is the distance fij ¼ 2pr ij between two interacting dislocations, l = 26 GPa is the shear modulus and b = 0.3 nm is the magnitude of the Burgers vector; when a dislocation is moving into a region within a distance of cut-off radius from a free surface, the image force is also worked out by this relation. (ii) A resistance stress opposite to the glide direction due to Taylor-type forest interactions p [33,34], which can be ffiffiffi written in the form of siforest ¼ alb q, where q is mobile dislocation density and a is a constant. (iii) An external stress linearly increasing with time applied to the simulation region normal to the radial direction. The resultant force decomposed along the slip direction si can be calculated, and the velocity of a mobile dislocation vi can be worked out by the simplified law v = ks. Sources resembling the Frank–Read type are randomly distributed in the simulation region with a given density qFRS. Each of them will probably nucleate a dislocation dipole when subjected to a resultant stress larger than a critical value rnuc continuously over 50 ms. The value of rnuc for these nucleation sources follows a Gaussian distribution, with a mean stress of 15 MPa and a standard deviation of 6 MPa. The spacing of the nucleated dislo1 lb ; so that the nucleated dipole cation dipole is Lnuc ¼ 2p rnuc cannot annihilate immediately by its own attraction force. Each dislocation source can only operate once until it has relaxed for a long enough time (250 ms). Precipitate particles are randomly placed in the simulation region. The interactions between dislocations and precipitates are various and complex [35], due to the different precipitate geometries and physical/chemical properties; therefore, for simplicity, we only consider particle shearing and cross slip in this work. The distance between two neighboring precipitates on a slip plane is supposed to obey a Gaussian distribution, with average spacing lave and standard deviation 0.3lave, so that the probability of meeting an obstacle when a dislocation has freely moved a distance d can be calculated as the accumulated integral of the Gaussian distribution. Once a dislocation meets a precipitate, if the resultant stress along the glide plane si is larger than the critical shear stress rc, then the dislocation can overcome the precipitate and keep on moving in the slip plane in a slower speed of vi = k(si rc). Otherwise, if the resultant stress normal to the slip direction is larger than another critical stress rcrs(rcrs = 3rc), the dislocation has a chance (of 10%) to cross slip over the particle. If the above conditions are not met, then the dislocation is considered to be pinned up by the precipitate. In each time step, a dislocation would move a distance of di = vi Dt, so the total strain in this time step can be worked out according P a to the Orowan equation as vi Dt, where A is the area De ¼ qbv Dt ¼ ðb=AÞ Ni¼1 of the simulation region and Na is the total number of active dislocations in the current time step. In the simulation, the value k = 0.72 lm s1 (MPa)1 in the velocity law is assumed, the time step Dt is set at 1 ms and the
loading rate r_ is 20 MPas1. The width of the simulation region is varied from 1.5 to 6.5 lm, while the lengths are maintained at three times the width. The average spacing between precipitates lave and the critical resistance by precipitates rc are varied to study their effects on the stress–strain behavior. The initial dislocation density q0 and the dislocation source density qFRS for pure Al micropillars are assumed to be 2.2 1012 and 1.1 1012 m2, respectively, while for the duralumin case, both q0 and qFRS are doubled, to 4.4 1012 and 2.2 1012m2, as the presence of precipitates may introduce more defects into the matrix. Fig. 1(a) shows typical simulated stress–strain curves of pure aluminum pillars with varied size, where the diameters of the micropillars are distinguished by different colors. These results are broadly in accordance with experimental observation [8,21], namely, the strength and deformation curves depend on the size of the specimen – the smallest 1.5 lm pillars exhibit the steepest stress–strain curves with successive fluctuations and the greatest scatter between different simulation runs, while the largest 6.5 lm pillars show the softest and smoothest stress–strain behavior, with less scattering as well. The simulated 2% proof strength of pure aluminum micropillars is found to approximately obey a power-law size dependence, as shown in Fig. 1(b). The simulation data are very close to the experimental observation, and the exponent m in the power-law relation r2% /Dm is about m = 0.79 ± 0.07, while experiment data [21] show m = 0.89 ± 0.07. Fig. 2(a and b) shows the simulated stress–strain behaviors of duralumin micropillars, when the average spacings of precipitate particles lave are 3 and 10 nm, respectively, and in both cases the critical shear stress rc is 51.8 MPa. Both groups exhibit much higher strength than the pure Al case, and an inverse relation between size and strength is still very significant. The
Fig. 1. (a) Simulated stress–strain behaviors of pure Al micropillars and their (b) 2% proof strength vs. diameter relation, in comparison with experimental data [21].
R. Gu et al. / Scripta Materialia 76 (2014) 73–76
Fig. 2. Simulated stress–strain behaviors of duralumin micropillars with mean precipitate spacing of (a) 3 nm and (b) 10 nm, and (c) their 2% proof strength with varying precipitate spacing.
stress–strain curves with lave = 3 nm (Fig. 2(a)) show that, when the pillar size is larger than 3.5 lm, the three stress–strain curves of the same size on repeated simulation trials are relatively smooth and coincident, with very little scattering, while the curves for smaller sizes are much more widely scattered and fluctuating. When lave is increased to 10 nm (Fig. 2(b)), the stress–strain curves for pillars with diameters of 1.5–6.5 lm are all jerky, with occasional strain bursts occurring even for a pillar size of 6.5 lm. It is suggested that, as the precipitate spacing decreases, mobile dislocations in a large pillar have a greater chance of meeting precipitates, which may resist and slow down the dislocations before they glide to free surfaces, so that their deformation curves are smoother and more stable. Fig. 2(c) shows the relation between the pillar strength and precipitate spacing. As lave varies from 3 to 20 nm, the 2% proof strength of pillars of the same size does not exhibit any significant correlation with lave. However, as lave increases beyond 30 nm, small pillars of 1.5–2.0 lm diameter can show a much higher strength on increasing lave, while pillars larger than 3.5 lm do not exhibit such a trend and their strength remains at almost the same value. This reveals that small-sized specimens are more sensitive to a change in the intrinsic structure and the size dependence of strength may be intensified as the internal length scale increases.
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Fig. 3 shows the simulated 2% proof strength r2% vs. diameter of pillars with precipitates of different critical shear stress rc when spacing lave is a constant 3 nm, together with the experimental data for peak-aged duralumin case [23]. It is obvious that the strength of duralumin micropillars exhibits a positive correlation with rc, and that their r2% still shows power-law scaling with diameter D as r2% /Dm, where m is 0.37 ± 0.04 and 0.51 ± 0.04, respectively, for rc values of 25.9 and 51.8 MPa, indicating that the size dependence of strength is weakened with the presence of precipitate particles. It is also evident that the value of rc = 51.8 MPa can fit the experimental data very well. Fig. 4(a and b) shows the variation in dislocation density inside the specimen with applied stress. For both the pure Al and duralumin micropillars of diameter 6.5 lm, their dislocation density initially increases with applied stress to a plateau, then decreases with further increasing stress, and the pure Al pillars exhibit the lowest dislocation density compared with the ones with precipitates, in goof agreement with experimental observations [23]. Although pillars with different precipitates show varied peak values of dislocation density, the plateau is very close to the point when the plastic strain starts to increase rapidly with stress, as shown in Fig. 1(a) and Fig. 2(a and b). The results here show that dislocations can initially accumulate and multiply quickly in the specimen with increasing applied stress, and the precipitates may resist the movement of dislocations until the applied stress is large enough to overcome the critical shear stress of the precipitates rc. The dislocations will then keep on gliding and deplete at free surfaces, so that the number of dislocations in the specimen will decrease, and even strain bursts will appear on the stress–strain curve. Fig. 4(a) also reveals that pillars with precipitates of large rc and dense spacing can show a high yield strength and a peak value of dislocation density, because in this case the precipitates are more capable and have a high probability of resisting the mobile dislocations, making them stay inside the confined volume. For the 1.5 lm case shown in Fig. 4(b), the dislocation density vs. stress curves fluctuate quite severely but still exhibit the above-mentioned trend, except for the case of pure Al pillars, due to the dramatic fluctuations. Because these pillars are so small, even the nucleation or annihilation of a single dislocation would significantly change the average number of active dislocations in the specimen. This simulation study shows that duralumin micropillars exhibit much higher strength than pure Al
Fig. 3. 2% Proof strength vs. diameter of duralumin micropillars with mean precipitate spacing of 3 nm, in comparison with experimental data [23].
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the precipitates. A fine microstructure can weaken the size dependence of strength and smooth the stress–strain behavior of duralumin micropillars. This work is supported by funding from Kingboard Endowed Professorship in Materials Engineering.
Fig. 4. Dislocation density of duralumin (a) 6.5 lm and (b) 1.5 lm pillars varied with applied stress in the simulation.
micropillars, while their size dependence of strength is much milder, and all these are comparable to our previous experimental observations [23]. The strength as well as spacing of the precipitates can significantly affect the stress–strain behavior of the micropillars. Fig. 2 shows that, when the precipitate spacing lave rises from 3 to 20 nm, the pillar strength does not vary significantly, but the stress–strain curves of big pillars 5–6 lm in size become rough and jerky. As the precipitate spacing lave increases to 30–40 nm, small pillars 1–2 lm in diameter exhibit even higher strength with increasing lave but the bigger pillars do not show any significant change; as a result, the exponent m in the scaling law r2% / Dm becomes larger as lave is increased to 30–40 nm. Previous experimental results [23] show a similar behavior, namely, duralumin pillars aged at the peak condition in which the precipitates are larger but more sparsely distributed are found to exhibit a stronger dependence of strength on size compared to pillars aged at room temperature, where the precipitates are smaller in both size and spacing. It is suggested that, in a very small crystal, as the precipitate spacing increases, a mobile dislocation may encounter only a few obstacles, so that it has a large chance to glide to a free surface, leaving the confined volume in a starved state, as shown in Fig. 4(b), where the dislocation number decreases significantly when the applied stress is 150 MPa and plastic strain is only 1%. The Orowan equation e_ ¼ qbv suggests that, as the dislocation density decreases, a higher stress is required to make the limited number of mobile dislocations slip faster so as to maintain a given strain rate. It is thus not surprising that small pillars with relatively sparsely distributed precipitates can exhibit even higher strength, and the size dependence of strength can be enhanced. To conclude, the present simulations show that precipitate hardened duralumin micropillars exhibit power-law size-dependent strength, which is mainly determined by the strength and dispersion distance of
[1] D.M. Dimiduk, M.D. Uchic, T.A. Parthasarathy, Acta Mater. 53 (2005) 4065. [2] R. Dou, B. Derby, Scripta Mater. 61 (2009) 524. [3] J.R. Greer, J.T.M. De Hosson, Prog. Mater Sci. 56 (2011) 654. [4] M.D. Uchic, D.M. Dimiduk, J.N. Florando, W.D. Nix, Science 305 (2004) 986. [5] D.M. Dimiduk, C. Woodward, R. LeSar, M.D. Uchic, Science 312 (2006) 1188. [6] J.R. Greer, W.D. Nix, Phys. Rev. B 73 (2006). [7] D.M. Dimiduk, M.D. Uchic, S.I. Rao, C. Woodward, T.A. Parthasarathy, Modell. Simul. Mater. Sci. Eng. 15 (2007) 135. [8] K.S. Ng, A.H.W. Ngan, Acta Mater. 56 (2008) 1712. [9] Z.W. Shan, R.K. Mishra, S.A.S. Asif, O.L. Warren, A.M. Minor, Nat. Mater. 7 (2008) 115. [10] S.H. Oh, M. Legros, D. Kiener, G. Dehm, Nat. Mater. 8 (2009) 95. [11] R. Maass, M.D. Uchic, Acta Mater. 60 (2012) 1027. [12] R. Maass, S. Van Petegem, D. Grolimund, H. Van Swygenhoven, M.D. Uchic, Appl. Phys. Lett. 91 (2007) 131909. [13] J.R. Greer, W.C. Oliver, W.D. Nix, Acta Mater. 53 (2005) 1821. [14] T.A. Parthasarathy, S.I. Rao, D.M. Dimiduk, M.D. Uchic, D.R. Trinkle, Scripta Mater. 56 (2007) 313. [15] S.I. Rao, D.M. Dimiduk, M. Tang, T.A. Parthasarathy, M.D. Uchic, C. Woodward, Philos. Mag. 87 (2007) 4777. [16] D.M. Norfleet, D.M. Dimiduk, S.J. Polasik, M.D. Uchic, M.J. Mills, Acta Mater. 56 (2008) 2988. [17] S. Rao, D. Dimiduk, T. Parthasarathy, M. Uchic, M. Tang, C. Woodward, Acta Mater. 56 (2008) 3245. [18] R. Gu, A.H.W. Ngan, J. Mech. Phys. Solids 61 (2013) 1531. [19] K.S. Ng, A.H.W. Ngan, Acta Mater. 57 (2009) 4902. [20] A.T. Jennings, C. Gross, F. Greer, Z.H. Aitken, S.W. Lee, C.R. Weinberger, J.R. Greer, Acta Mater. 60 (2012) 3444. [21] R. Gu, A.H.W. Ngan, Acta Mater. 60 (2012) 6102. [22] D. Jang, C. Cai, J.R. Greer, Nano Lett. 11 (2011) 1743. [23] R. Gu, A.H.W. Ngan, Scripta Mater. 68 (2013) 861. [24] K.S. Ng, A.H.W. Ngan, Philos. Mag. 89 (2009) 3013. [25] A. Kunz, S. Pathak, J.R. Greer, Acta Mater. 59 (2011) 4416. [26] A. Rinaldi, P. Peralta, C. Friesen, K. Sieradzki, Acta Mater. 56 (2008) 511. [27] X.W. Gu, C.N. Loynachan, Z. Wu, Y.-W. Zhang, D.J. Srolovitz, J.R. Greer, Nano Lett. 12 (2012) 6385. [28] K.A. Afanasyev, F. Sansoz, Nano Lett. 7 (2007) 2056. [29] D. Jang, X. Li, H. Gao, J.R. Greer, Nat. Nanotechnol. 7 (2012) 594. [30] L. Nicola, E. Van der Giessen, A. Needleman, J. Appl. Phys. 93 (2003) 5920. [31] Z. Li, C. Hou, M. Huang, C. Ouyang, Comput. Mater. Sci. 46 (2009) 1124. [32] R. Kumar, L. Nicola, E. Van der Giessen, Mater. Sci. Eng. A 527 (2009) 7. [33] P.S.S. Leung, A.H.W Ngan, Scripta Mater 69 (2013) 235. [34] A.A. Benzerga, J. Mech. Phys. Solids 57 (2009) 1459. [35] J.W. Martin, Precipitation Hardening, ButterworthHeinemann, Boston, MA, 1998.
ScienceDirect Scripta Materialia 76 (2014) 73–76 www.elsevier.com/locate/scriptamat
Size effect on deformation of duralumin micropillars – A dislocation dynamics study R. Gu,⇑ P.P.S. Leung and A.H.W. Ngan Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China Received 7 November 2013; accepted 26 December 2013 Available online 3 January 2014
Two-dimensional dislocation dynamics simulations are used to study precipitate strengthening effects in duralumin micropillars. The results show that a refined microstructure may resist and slow down the movement of dislocations inside the confined volume, leading to hardening and a weak dependence of strength on size. This study illustrates that the deformation behavior of small crystals is controlled by the combined effect of the internal length scale and the external size of the crystal. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Plastic deformation; Dislocation dynamics; Size effect; Precipitate hardening
The effect of size on the deformation behavior of metallic materials has been intensively studied in recent years. One of the most significant findings is that metallic single crystals show a power-law size dependence of strength [1–3] and jerky stress–strain behavior in a stochastic manner [1,4–9]. With evidences gained from in situ transmission electron microscopy [9,10] and Laue microdiffraction [11,12], a number of physical models have been devised to explain the deformation behavior of small crystals. The “dislocation starvation” [6,13] theory considers that mobile dislocations in a small crystal would easily slip through the small crystal without accumulating and multiplying, so that the crystal is constantly maintained in a dislocation-free state. Another group of models consider that the small crystal is not dislocation free initially, but the operation of dislocation sources [14,15] and the interaction of dislocations in a confined environment lacking the mean field-averaging effect [16,17] may result in unusual deformation behavior. Another model [18] explicitly predicts that a power-law size dependence of strength stems from Taylor-type interactions in a fractal dislocation network, and that the power-law exponent is inversely related to the fractal dimension of the dislocation network. In addition to the extrinsic size effect, several recent studies have explored how the mechanical properties of small crystals are affected by changes in the intrinsic
⇑ Corresponding author. Tel.: +852 28592624. e-mail: raicy4197@ gmail.com
microstructural length scale, such as decreasing the interaction distances between dislocations by trapping them inside the crystal by surface coating [19–22], or introducing precipitates [23], grain boundaries [24,25], or nanocrystalline [26,27] or nanotwinning [22,28,29] microstructures into the crystal. Our previous experimental study [23] on the deformation behavior of duralumin micropillars has shown that, with the presence of second-phase precipitates, strength is much improved and less size-dependent, and a very high density of dislocation debris is retained in the deformed microstructure, suggesting that mobile dislocations are slowed down by viscous drag from the precipitates. In this study, two-dimensional (2-D) dislocation dynamics simulations [30–33] are used to study the deformation behavior of such duralumin micropillars. The distribution of the precipitate particles and the external size of the specimen are varied to investigate their effects on the plastic properties of the micropillars. In the present 2-D simulations, screw dislocations are considered, and these are randomly distributed in a simulation region with a given initial density q0. The simulation region is rectangular, with the short and long sides (in a ratio of 1:3) respectively representing surfaces parallel and perpendicular to the radial direction of the micropillar. Periodic boundary conditions are applied at the two short sides of the simulation region, meaning that as a dislocation escapes from one side it re-enters into the region from the opposite side. The other two (long) sides are regarded as free surfaces where nearby dislocations would be affected by image forces and
1359-6462/$ - see front matter Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.scriptamat.2013.12.020
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may therefore likely be depleted on them. Two orthogonal slip systems are considered in the simulation region, which are oriented at 35.2 and 125.2° from the radial direction. The material properties of aluminum are used in this simulation. Dislocations gliding in the slip plane may sustain three types of forces without meeting any precipitates: (i) the pairwise elastic interaction force lb2 between dislocations, where rij is the distance fij ¼ 2pr ij between two interacting dislocations, l = 26 GPa is the shear modulus and b = 0.3 nm is the magnitude of the Burgers vector; when a dislocation is moving into a region within a distance of cut-off radius from a free surface, the image force is also worked out by this relation. (ii) A resistance stress opposite to the glide direction due to Taylor-type forest interactions p [33,34], which can be ffiffiffi written in the form of siforest ¼ alb q, where q is mobile dislocation density and a is a constant. (iii) An external stress linearly increasing with time applied to the simulation region normal to the radial direction. The resultant force decomposed along the slip direction si can be calculated, and the velocity of a mobile dislocation vi can be worked out by the simplified law v = ks. Sources resembling the Frank–Read type are randomly distributed in the simulation region with a given density qFRS. Each of them will probably nucleate a dislocation dipole when subjected to a resultant stress larger than a critical value rnuc continuously over 50 ms. The value of rnuc for these nucleation sources follows a Gaussian distribution, with a mean stress of 15 MPa and a standard deviation of 6 MPa. The spacing of the nucleated dislo1 lb ; so that the nucleated dipole cation dipole is Lnuc ¼ 2p rnuc cannot annihilate immediately by its own attraction force. Each dislocation source can only operate once until it has relaxed for a long enough time (250 ms). Precipitate particles are randomly placed in the simulation region. The interactions between dislocations and precipitates are various and complex [35], due to the different precipitate geometries and physical/chemical properties; therefore, for simplicity, we only consider particle shearing and cross slip in this work. The distance between two neighboring precipitates on a slip plane is supposed to obey a Gaussian distribution, with average spacing lave and standard deviation 0.3lave, so that the probability of meeting an obstacle when a dislocation has freely moved a distance d can be calculated as the accumulated integral of the Gaussian distribution. Once a dislocation meets a precipitate, if the resultant stress along the glide plane si is larger than the critical shear stress rc, then the dislocation can overcome the precipitate and keep on moving in the slip plane in a slower speed of vi = k(si rc). Otherwise, if the resultant stress normal to the slip direction is larger than another critical stress rcrs(rcrs = 3rc), the dislocation has a chance (of 10%) to cross slip over the particle. If the above conditions are not met, then the dislocation is considered to be pinned up by the precipitate. In each time step, a dislocation would move a distance of di = vi Dt, so the total strain in this time step can be worked out according P a to the Orowan equation as vi Dt, where A is the area De ¼ qbv Dt ¼ ðb=AÞ Ni¼1 of the simulation region and Na is the total number of active dislocations in the current time step. In the simulation, the value k = 0.72 lm s1 (MPa)1 in the velocity law is assumed, the time step Dt is set at 1 ms and the
loading rate r_ is 20 MPas1. The width of the simulation region is varied from 1.5 to 6.5 lm, while the lengths are maintained at three times the width. The average spacing between precipitates lave and the critical resistance by precipitates rc are varied to study their effects on the stress–strain behavior. The initial dislocation density q0 and the dislocation source density qFRS for pure Al micropillars are assumed to be 2.2 1012 and 1.1 1012 m2, respectively, while for the duralumin case, both q0 and qFRS are doubled, to 4.4 1012 and 2.2 1012m2, as the presence of precipitates may introduce more defects into the matrix. Fig. 1(a) shows typical simulated stress–strain curves of pure aluminum pillars with varied size, where the diameters of the micropillars are distinguished by different colors. These results are broadly in accordance with experimental observation [8,21], namely, the strength and deformation curves depend on the size of the specimen – the smallest 1.5 lm pillars exhibit the steepest stress–strain curves with successive fluctuations and the greatest scatter between different simulation runs, while the largest 6.5 lm pillars show the softest and smoothest stress–strain behavior, with less scattering as well. The simulated 2% proof strength of pure aluminum micropillars is found to approximately obey a power-law size dependence, as shown in Fig. 1(b). The simulation data are very close to the experimental observation, and the exponent m in the power-law relation r2% /Dm is about m = 0.79 ± 0.07, while experiment data [21] show m = 0.89 ± 0.07. Fig. 2(a and b) shows the simulated stress–strain behaviors of duralumin micropillars, when the average spacings of precipitate particles lave are 3 and 10 nm, respectively, and in both cases the critical shear stress rc is 51.8 MPa. Both groups exhibit much higher strength than the pure Al case, and an inverse relation between size and strength is still very significant. The
Fig. 1. (a) Simulated stress–strain behaviors of pure Al micropillars and their (b) 2% proof strength vs. diameter relation, in comparison with experimental data [21].
R. Gu et al. / Scripta Materialia 76 (2014) 73–76
Fig. 2. Simulated stress–strain behaviors of duralumin micropillars with mean precipitate spacing of (a) 3 nm and (b) 10 nm, and (c) their 2% proof strength with varying precipitate spacing.
stress–strain curves with lave = 3 nm (Fig. 2(a)) show that, when the pillar size is larger than 3.5 lm, the three stress–strain curves of the same size on repeated simulation trials are relatively smooth and coincident, with very little scattering, while the curves for smaller sizes are much more widely scattered and fluctuating. When lave is increased to 10 nm (Fig. 2(b)), the stress–strain curves for pillars with diameters of 1.5–6.5 lm are all jerky, with occasional strain bursts occurring even for a pillar size of 6.5 lm. It is suggested that, as the precipitate spacing decreases, mobile dislocations in a large pillar have a greater chance of meeting precipitates, which may resist and slow down the dislocations before they glide to free surfaces, so that their deformation curves are smoother and more stable. Fig. 2(c) shows the relation between the pillar strength and precipitate spacing. As lave varies from 3 to 20 nm, the 2% proof strength of pillars of the same size does not exhibit any significant correlation with lave. However, as lave increases beyond 30 nm, small pillars of 1.5–2.0 lm diameter can show a much higher strength on increasing lave, while pillars larger than 3.5 lm do not exhibit such a trend and their strength remains at almost the same value. This reveals that small-sized specimens are more sensitive to a change in the intrinsic structure and the size dependence of strength may be intensified as the internal length scale increases.
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Fig. 3 shows the simulated 2% proof strength r2% vs. diameter of pillars with precipitates of different critical shear stress rc when spacing lave is a constant 3 nm, together with the experimental data for peak-aged duralumin case [23]. It is obvious that the strength of duralumin micropillars exhibits a positive correlation with rc, and that their r2% still shows power-law scaling with diameter D as r2% /Dm, where m is 0.37 ± 0.04 and 0.51 ± 0.04, respectively, for rc values of 25.9 and 51.8 MPa, indicating that the size dependence of strength is weakened with the presence of precipitate particles. It is also evident that the value of rc = 51.8 MPa can fit the experimental data very well. Fig. 4(a and b) shows the variation in dislocation density inside the specimen with applied stress. For both the pure Al and duralumin micropillars of diameter 6.5 lm, their dislocation density initially increases with applied stress to a plateau, then decreases with further increasing stress, and the pure Al pillars exhibit the lowest dislocation density compared with the ones with precipitates, in goof agreement with experimental observations [23]. Although pillars with different precipitates show varied peak values of dislocation density, the plateau is very close to the point when the plastic strain starts to increase rapidly with stress, as shown in Fig. 1(a) and Fig. 2(a and b). The results here show that dislocations can initially accumulate and multiply quickly in the specimen with increasing applied stress, and the precipitates may resist the movement of dislocations until the applied stress is large enough to overcome the critical shear stress of the precipitates rc. The dislocations will then keep on gliding and deplete at free surfaces, so that the number of dislocations in the specimen will decrease, and even strain bursts will appear on the stress–strain curve. Fig. 4(a) also reveals that pillars with precipitates of large rc and dense spacing can show a high yield strength and a peak value of dislocation density, because in this case the precipitates are more capable and have a high probability of resisting the mobile dislocations, making them stay inside the confined volume. For the 1.5 lm case shown in Fig. 4(b), the dislocation density vs. stress curves fluctuate quite severely but still exhibit the above-mentioned trend, except for the case of pure Al pillars, due to the dramatic fluctuations. Because these pillars are so small, even the nucleation or annihilation of a single dislocation would significantly change the average number of active dislocations in the specimen. This simulation study shows that duralumin micropillars exhibit much higher strength than pure Al
Fig. 3. 2% Proof strength vs. diameter of duralumin micropillars with mean precipitate spacing of 3 nm, in comparison with experimental data [23].
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the precipitates. A fine microstructure can weaken the size dependence of strength and smooth the stress–strain behavior of duralumin micropillars. This work is supported by funding from Kingboard Endowed Professorship in Materials Engineering.
Fig. 4. Dislocation density of duralumin (a) 6.5 lm and (b) 1.5 lm pillars varied with applied stress in the simulation.
micropillars, while their size dependence of strength is much milder, and all these are comparable to our previous experimental observations [23]. The strength as well as spacing of the precipitates can significantly affect the stress–strain behavior of the micropillars. Fig. 2 shows that, when the precipitate spacing lave rises from 3 to 20 nm, the pillar strength does not vary significantly, but the stress–strain curves of big pillars 5–6 lm in size become rough and jerky. As the precipitate spacing lave increases to 30–40 nm, small pillars 1–2 lm in diameter exhibit even higher strength with increasing lave but the bigger pillars do not show any significant change; as a result, the exponent m in the scaling law r2% / Dm becomes larger as lave is increased to 30–40 nm. Previous experimental results [23] show a similar behavior, namely, duralumin pillars aged at the peak condition in which the precipitates are larger but more sparsely distributed are found to exhibit a stronger dependence of strength on size compared to pillars aged at room temperature, where the precipitates are smaller in both size and spacing. It is suggested that, in a very small crystal, as the precipitate spacing increases, a mobile dislocation may encounter only a few obstacles, so that it has a large chance to glide to a free surface, leaving the confined volume in a starved state, as shown in Fig. 4(b), where the dislocation number decreases significantly when the applied stress is 150 MPa and plastic strain is only 1%. The Orowan equation e_ ¼ qbv suggests that, as the dislocation density decreases, a higher stress is required to make the limited number of mobile dislocations slip faster so as to maintain a given strain rate. It is thus not surprising that small pillars with relatively sparsely distributed precipitates can exhibit even higher strength, and the size dependence of strength can be enhanced. To conclude, the present simulations show that precipitate hardened duralumin micropillars exhibit power-law size-dependent strength, which is mainly determined by the strength and dispersion distance of
[1] D.M. Dimiduk, M.D. Uchic, T.A. Parthasarathy, Acta Mater. 53 (2005) 4065. [2] R. Dou, B. Derby, Scripta Mater. 61 (2009) 524. [3] J.R. Greer, J.T.M. De Hosson, Prog. Mater Sci. 56 (2011) 654. [4] M.D. Uchic, D.M. Dimiduk, J.N. Florando, W.D. Nix, Science 305 (2004) 986. [5] D.M. Dimiduk, C. Woodward, R. LeSar, M.D. Uchic, Science 312 (2006) 1188. [6] J.R. Greer, W.D. Nix, Phys. Rev. B 73 (2006). [7] D.M. Dimiduk, M.D. Uchic, S.I. Rao, C. Woodward, T.A. Parthasarathy, Modell. Simul. Mater. Sci. Eng. 15 (2007) 135. [8] K.S. Ng, A.H.W. Ngan, Acta Mater. 56 (2008) 1712. [9] Z.W. Shan, R.K. Mishra, S.A.S. Asif, O.L. Warren, A.M. Minor, Nat. Mater. 7 (2008) 115. [10] S.H. Oh, M. Legros, D. Kiener, G. Dehm, Nat. Mater. 8 (2009) 95. [11] R. Maass, M.D. Uchic, Acta Mater. 60 (2012) 1027. [12] R. Maass, S. Van Petegem, D. Grolimund, H. Van Swygenhoven, M.D. Uchic, Appl. Phys. Lett. 91 (2007) 131909. [13] J.R. Greer, W.C. Oliver, W.D. Nix, Acta Mater. 53 (2005) 1821. [14] T.A. Parthasarathy, S.I. Rao, D.M. Dimiduk, M.D. Uchic, D.R. Trinkle, Scripta Mater. 56 (2007) 313. [15] S.I. Rao, D.M. Dimiduk, M. Tang, T.A. Parthasarathy, M.D. Uchic, C. Woodward, Philos. Mag. 87 (2007) 4777. [16] D.M. Norfleet, D.M. Dimiduk, S.J. Polasik, M.D. Uchic, M.J. Mills, Acta Mater. 56 (2008) 2988. [17] S. Rao, D. Dimiduk, T. Parthasarathy, M. Uchic, M. Tang, C. Woodward, Acta Mater. 56 (2008) 3245. [18] R. Gu, A.H.W. Ngan, J. Mech. Phys. Solids 61 (2013) 1531. [19] K.S. Ng, A.H.W. Ngan, Acta Mater. 57 (2009) 4902. [20] A.T. Jennings, C. Gross, F. Greer, Z.H. Aitken, S.W. Lee, C.R. Weinberger, J.R. Greer, Acta Mater. 60 (2012) 3444. [21] R. Gu, A.H.W. Ngan, Acta Mater. 60 (2012) 6102. [22] D. Jang, C. Cai, J.R. Greer, Nano Lett. 11 (2011) 1743. [23] R. Gu, A.H.W. Ngan, Scripta Mater. 68 (2013) 861. [24] K.S. Ng, A.H.W. Ngan, Philos. Mag. 89 (2009) 3013. [25] A. Kunz, S. Pathak, J.R. Greer, Acta Mater. 59 (2011) 4416. [26] A. Rinaldi, P. Peralta, C. Friesen, K. Sieradzki, Acta Mater. 56 (2008) 511. [27] X.W. Gu, C.N. Loynachan, Z. Wu, Y.-W. Zhang, D.J. Srolovitz, J.R. Greer, Nano Lett. 12 (2012) 6385. [28] K.A. Afanasyev, F. Sansoz, Nano Lett. 7 (2007) 2056. [29] D. Jang, X. Li, H. Gao, J.R. Greer, Nat. Nanotechnol. 7 (2012) 594. [30] L. Nicola, E. Van der Giessen, A. Needleman, J. Appl. Phys. 93 (2003) 5920. [31] Z. Li, C. Hou, M. Huang, C. Ouyang, Comput. Mater. Sci. 46 (2009) 1124. [32] R. Kumar, L. Nicola, E. Van der Giessen, Mater. Sci. Eng. A 527 (2009) 7. [33] P.S.S. Leung, A.H.W Ngan, Scripta Mater 69 (2013) 235. [34] A.A. Benzerga, J. Mech. Phys. Solids 57 (2009) 1459. [35] J.W. Martin, Precipitation Hardening, ButterworthHeinemann, Boston, MA, 1998.